Problem: Here's a partially-filled Hessian matrix. $\begin{bmatrix} 2 & ??? \\ \\ 3e^{3y} & 9xe^{3y} \end{bmatrix}$ What is the missing entry? Choose 1 answer: Choose 1 answer: (Choice A) A $9e^{3y}$ (Choice B) B $xe^{3y}$ (Choice C) C $3e^{3y}$ (Choice D) D There's not enough information.
Solution: The Hessian of a scalar field $f$ is the matrix that contains all its second-order partial derivative information. $\bold{H}(f) = \begin{bmatrix} f_{xx} & f_{xy} \\ \\ f_{yx} & f_{yy} \end{bmatrix}$ Because the order of mixed partial derivatives often doesn't matter, the bottom left and top right entries of the Hessian are usually the same. Matching to the bottom left corner of the matrix, the missing entry is therefore $3e^{3y}$.